Download Dualities for Intuitionistic Modal Logics

Dualities for Intuitionistic Modal Logics
Alessandra Palmigiano∗
Abstract
We present a duality for the intuitionistic modal logic IK introduced by Fischer Servi in
[10, 11]. Unlike other dualities for IK, the dual structures of the duality presented here are
ordered topological spaces endowed with just one extra relation, that is used to define the
set-theoretic representation of both 2 and 3. We also give a parallel presentation of dualities
for the intuitionistic modal logics I2 and I3 . Finally, we turn to the intuitionistic modal logic
MIPC, and give a very natural characterization of the dual spaces for MIPC introduced in [2]
as a subcategory of the category of the dual spaces for IK introduced here.
1
Introduction and Preliminaries
Fischer Servi [10] proposed a general method for defining the intuitionistic analogue of a given
classical modal system. Her method is based on a translation of formulas of the intuitionistic
modal language L23 = {∧, ∨, →, ¬, 2, 3} into a classical modal language with two necessity
operators, that extends the Gödel translation of the intuitionistic propositional calculus IPC into
Lewis’ classical modal logic S4. The intuitionistic modal logics K-IC (IK elsewhere) and S5-IC
(I5 elsewhere) in the language L23 are defined in the same paper according to this method as the
intuitionistic counterparts of the classical modal logics K and S5, and are axiomatized in [11]. In
particular I5, which coincides (see also [12]) with Prior’s intuitionistic modal logic MIPC [16], is
axiomatized by extending the set of axioms of IK with the following ones:
2p → p
p → 3p
2p → 22p
33p → 3p
3p → 23p
32p → 2p,
which shows that MIPC is to IK what S5 is to K, also from the point of view of axioms. A distinguished feature of IK is that the modal operators are not interdefinable but they are connected
by the pair of “connecting axioms”(see (1) in subsection 1.1): In this respect, IK is similar to the
negation- and implication-free modal logic PML [7], for which a duality was established in [3],
relating PML-algebras with structures consisting of ordered topological spaces (indeed, Priestley
spaces [6]) endowed with one extra relation, and called K+ -spaces. This paper presents a “duality
for IK”, i.e. a duality relating algebraic and topological semantics of IK. Like the duality in [3] and
unlike other dualities for IK reported in the literature (see for example [17]), the dual structures
of the duality presented here are ordered topological spaces (indeed, Esakia spaces [9], see also
[5, 14]) endowed with just one extra relation, that is used to define the set-theoretic representation
of both 2 and 3. Also, unlike the duality in [17], this duality naturally extends the definitions
and techniques used by Fischer Servi in the proof of completeness for IK via canonical model
construction [12]. The similarities between the duality presented here and the one in [3] confirm
the intuition that PML and IK are akin, and indeed, the motivation of this work came from the
project of extending to IK the results on PML presented in [14], where an equivalence of categories was established between K+ -spaces and the coalgebras of the Vietoris functor on Priestley
spaces. This class of coalgebras is therefore as adequate a semantics for PML as the algebraic
one (consisting of PML-algebras), and as the topological/relational one (consisting of K+ -spaces),
∗ Partially supported by Catalan grant 2001FI 00281 UB PG, Spanish grant DGESIC BFM2001-3329, and
Catalan grant 2001SGR-00017.
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and this gives a very concrete sense to the expression “PML is a coalgebraic logic”, i.e. the logic
of a category of coalgebras. The duality that we are going to present is a basic ingredient in the
investigation on whether IK or MIPC can be declared coalgebraic logics as well.
In the remainder of this section, we are going to formally introduce the logic IK together with
the other two intuitionistic modal logics I2 and I3 , and the categories of algebras canonically
associated with each of them. In section 2, we define the frames of these logics, and show that
their associated complex algebras belong to the expected categories. In section 3 we define the
general frames and p-morphisms for these logics. In sections 4 and 5 we establish back-and-forth
functorial correspondences between general frames and algebras, and between p-morphisms and
algebra homomorphisms, for each logic. In section 6 we establish the dualities between the algebras
and suitable full subcategories of the general frames categories: For L ∈ {I2 , I3 , IK} the objects
of these subcategories are called L-spaces, and play an analogous role to descriptive general frames
for the classical modal logic K. In section 7 we characterize the topological semantics of MIPC
within the category of IK-spaces.
1.1
The logics
From now on, we take L = {∧, ∨, →, ⊥, >} as the intuitionistic propositional language, or equivalently, as the algebraic similarity type of Heyting algebras. For a non-empty set M of unary modal
operators, let LM be the intuitionistic propositional language augmented by the connectives in
M . By an intuitionistic modal logic we understand any subset of LM containing all the theorems
of IPC and closed under modus ponens, substitution and the regularity rule ϕ → ψ/mϕ → mψ
for every m ∈ M .
The logic I2 , in the language L2 , is axiomatized by adding the following axioms to IPC:
2(φ ∧ ψ) = 2φ ∧ 2ψ and 2> = >.
The logic I3 , in the language L3 , is axiomatized by adding the following axioms to IPC:
3(φ ∨ ψ) = 3φ ∨ 3ψ and 3⊥ = ⊥.
The logic I23 is the smallest logic S in the language L23 that contains I2 and I3 . IK is the
axiomatic extension of I23 obtained by adding the connecting axioms
3(φ → ψ) → (2φ → 3ψ) and
1.2
(3φ → 2ψ) → 2(φ → ψ).
(1)
Algebras for intuitionistic modal logics
An L2 -algebra A is an I2 -algebra if its L-reduct is a Heyting algebra and the following axioms are
satisfied:
2(a ∧ b) = 2a ∧ 2b and 21 = 1.
An L3 -algebra A is an I3 -algebra if its L-reduct is a Heyting algebra and the following axioms are
satisfied:
3(a ∨ b) = 3a ∨ 3b and 30 = 0.
An L23 -algebra A is an IK-algebra if its L-reduct is a Heyting algebra and the following axioms
are satisfied:
1. 21 = 1
3. 2(a ∧ b) = 2a ∧ 2b
5. 3(a → b) ≤ 2a → 3b
2. 30 = 0
4. 3(a ∨ b) = 3a ∨ 3b
6. 3a → 2b ≤ 2(a → b).
For L ∈ {I2 , I3 , IK}, let LAlg be the category of L-algebras and their homomorphisms. Clearly,
LAlg is closed under subalgebras.
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2
Frames
For every relation S ⊆ X × X and every Y ⊆ X, let
S[Y ] = {x ∈ X | ySx for some y ∈ Y } and S −1 [Y ] = {x ∈ X | xSy for some y ∈ Y }.
We abbreviate S[{x}] and S −1 [{x}] with S[x] and S −1 [x] respectively. A preorder is a structure
hX, ≤i, such that X 6= ∅ and ≤ is a reflexive and transitive binary relation on X. For every
preorder hX, ≤i and every Y ⊆ X, Y ↑ = ≤[Y ] and Y ↓ = ≤−1 [Y ]. Y is an up-set (a down-set) if
Y = Y ↑ (Y = Y ↓). We abbreviate {x}↑ and {x}↓ with x↑ and x↓ respectively. Up(X) (Down(X))
is the collection of the up-sets (down-sets) of X.
A map f : hX, ≤i −→ hY, ≤i between preorders is strongly isotone if the following conditions are
satisfied for every x, y ∈ X and every z ∈ Y :
M1. If x ≤ y then f (x) ≤ f (y).
M2. If f (x) ≤ z then f (x0 ) = z for some x0 ∈ x↑.
An intuitionistic frame [4] is a poset, i.e. an antisymmetric preorder. For every Y, Z ⊆ X,
3S (Y ) =
2S (Y ) =
Z ⇒S Y =
{x ∈ X | S[x] ∩ Y 6= ∅} = S −1 [Y ]
{x ∈ X | S[x] ⊆ Y } = X \ S −1 [X \ Y ]
{x ∈ X | S[x] ∩ Z ⊆ Y } = 2S ((X \ Z) ∪ Y ).
We will always abbreviate ⇒≤ with ⇒. One can easily verify that for every poset hX, ≤i and
every A, B, C ∈ Up(X), A ⇒ B ∈ Up(X) and (A ∩ C) ⊆ B iff C ⊆ (A ⇒ B), which implies that
hUp(X), ∩, ∪, ⇒, ∅, Xi is a Heyting algebra.
Definition 2.1. (Frames) Let F = hX, ≤, Ri be a relational structure such that ≤ is a preorder.
1. F is an I2 -frame iff (≤ ◦ R) ⊆ (R ◦ ≤).
2. F is an I3 -frame iff (≥ ◦ R) ⊆ (R ◦ ≥).
3. F is an IK-frame iff (≥ ◦ R) ⊆ (R ◦ ≥) and (R ◦ ≤) ⊆ (≤ ◦ R).
If hX, ≤i is a poset, then hX, ≤, ≤i is an I2 -frame, hX, ≤, ≥i is an I3 -frame, and hX, ≤, ≥ ◦ ≤i
is an IK-frame. The following lemma is easy to show by direct computation, and explains the
meaning of the conditions in the definition of frames in algebraic terms:
Lemma 2.2. For every preorder hX, ≤i and every binary relation S on X,
1. (≤ ◦ S) ⊆ (S ◦ ≤) iff Up(X) is closed under 2S .
2. (≥ ◦ S) ⊆ (S ◦ ≥) iff Up(X) is closed under 3S .
3. (S ◦ ≤) ⊆ (≤ ◦ S) iff S[x↑] ∈ Up(X) for every x ∈ X.
Notice that for every every preorder hX, ≤i and every binary relation R on X, the following
inclusion (≤ ◦ (≤ ◦ R)) ⊆ ((≤ ◦ R) ◦ ≤) always holds, hence by applying 2.2 (1) we get:
Corollary 2.3. For every preorder hX, ≤i and every binary relation R on X, Up(X) is closed
under 2(≤◦R) .
Proposition 2.4. Let F = hX, ≤, Ri be a relational structure.
1. If F is an I2 -frame, then AF = hUp(X), ∩, ∪, ⇒, 2R , ∅, Xi is an I2 -algebra. Hence, every
subalgebra A of AF is an I2 -algebra.
2. If F is an I3 -frame, then AF = hUp(X), ∩, ∪, ⇒, 3R , ∅, Xi is an I3 -algebra. Hence, every
subalgebra A of AF is an I3 -algebra.
3. If F is an IK-frame, then AF = hUp(X), ∩, ∪, ⇒, 2(≤◦R) , 3R , ∅, Xi is an IK-algebra. Hence,
every subalgebra A of AF is an IK-algebra.
Proof. We only check axioms 5 and 6 for IK-algebras: Let us show that for every U, V ∈ Up(X)
153
(a) 3R (U ⇒ V ) ⊆ (2(≤◦R) U ⇒ 3R V )
and
(b) (3R U ⇒ 2(≤◦R) V ) ⊆ 2(≤◦R) (U ⇒ V ).
(a) Assume that x ∈ 3R (U ⇒ V ), let x ≤ z and z ∈ 2(≤◦R) U , and let us show that z ∈ 3R V , i.e.
that R[z] ∩ V 6= ∅. As x ∈ 3R (U ⇒ V ), then there exists y ∈ R[x] ∩ (U ⇒ V ), hence z ≥ xRy,
and so, as F is an IK-frame, zRv ≥ y for some v ∈ X. As v ∈ R[z] ⊆ (≤ ◦ R)[z] ⊆ U , y ≤ v and
y ∈ (U ⇒ V ), then v ∈ V , and as v ∈ R[z], then R[z] ∩ V 6= ∅.
(b) Assume that x ∈ (3R U ⇒ 2(≤◦R) V ), let z ∈ (≤ ◦ R)[x] and z ≤ y ∈ U , and let us show
that y ∈ V . As z ∈ (≤ ◦ R)[x], then x ≤ vRz ≤ y for some v ∈ X, hence, as F is an IK-frame,
x ≤ v ≤ wRy for some w ∈ X. As wRy ∈ U , then w ∈ 3R U , and as x ≤ w, then w ∈ 2(≤◦R) V ,
hence y ∈ R[w] ⊆ (≤ ◦ R)[w] ⊆ Y .
3
Topological semantics
Definition 3.1. (General frame) A general frame is a structure G = hX, ≤, R, Ai such that
X is a nonempty set, ≤ is a partial order on X, R is a binary relation on X, and A is a
subalgebra of hUp(X), ∩, ∪, ⇒, ∅, Xi. For every general frame G, FG = hX, ≤, Ri is its associated
frame, and its associated ordered topological space XG = hX, ≤, τA i has the following subbase:
{Y | Y ∈ A} ∪ {(X \ Y ) | Y ∈ A}.
We recall that a Priestley space is a structure X = hX, ≤, τ i such that hX, ≤i is a partial order,
hX, τ i is a compact topological space which is totally order-disconnected, i.e. for every x, y ∈ X,
if x 6≤ y then x ∈ U and y ∈
/ U for some clopen up-set U . An Esakia space X = hX, ≤, τ i is a
Priestley space such that for every clopen subset U of X, U ↓ = {x ∈ X | x ≤ u for some u ∈ U }
is clopen. For every (preordered) topological space X, K(X) is the set of closed subsets of X, and
K ↑ (X) is the set of closed up-sets of X.
For sake of self-containment, we report the following basic fact on Priestley spaces, that will be
needed onwards:
Lemma 3.2. For every Priestley space X = hX, ≤, τ i and every F ∈ K(X), F ↑ and F ↓ are closed
subsets of X. Hence, for every x ∈ X, x↑ and x↓ are closed subsets of X.
Proof. In order to show that F ↑ ∈ K(X), assume that x ∈
/ F ↑, and show that x ∈ A and
A ∩ F ↑ = ∅ for some A ∈ τ .
If x ∈
/ F ↑, then for every y ∈ F , x ∈
/ y↑, i.e. y 6≤ x. Then by total order-disconnectedness,
for
S
every y ∈ F there exists a clopen up-set Uy such that y ∈ Uy and x ∈
/ Uy . Therefore F ⊆ y∈F Uy ,
Sn
and as F is compact, for F is a closed
Sn subset of the compact space X, then F ⊆ i=1 Uyi for
some y1 , . . . , yn ∈ F . Let A = X \ i=1 Uyi . A is an open down-set of X, x ∈ A and A ∩ F = ∅.
Let us show that A ∩ F ↑ = ∅. Suppose that z ∈ A ∩ F ↑ for some z ∈ X. Then z ∈ A and y0 ≤ z
for some y0 ∈ F , and as A is a down-set, then y0 ∈ A, hence y0 ∈ A ∩ F = ∅, contradiction. The
proof that F ↓ ∈ K(X) is similar. As for the second part of the statement, since X is Hausdorff,
then {x} is closed for every x ∈ X.
Next, we are going to define, for L ∈ {I2 , I3 , IK}, the categories LGF of general L-frames and
their p-morphisms.
3.3
General I2 -frames and their morphisms
Definition 3.4. (General I2 -frame) Let G = hX, ≤, R, Ai be a general frame. G is a general
I2 -frame iff the following list of conditions is satisfied:
D1. XG is an Esakia space, and A is the collection of the clopen up-sets of XG .
D2’. A is closed under 2R .
D3. For every x ∈ X, R[x] ∈ K(XG ).
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Definition 3.5. (p-morphism of general I2 -frames) Let Gi = hXi , ≤, Ri , Ai i be general I2 frames, i = 1, 2. A map f : X1 → X2 is a p-morphism iff the following list of conditions is
satisfied for every x, x0 , y ∈ X1 , z ∈ X2 :
M1. if x ≤ y then f (x) ≤ f (y).
M2. If f (x) ≤ z then f (x0 ) = z for some x0 ∈ x↑.
M3. For every Y ∈ A2 , f −1 [Y ] ∈ A1 .
M4. If xR1 y then f (x)R2 f (y).
M5’. If f (x)R2 z then f (x0 ) ≤ z for some x0 ∈ R1 [x].
By M1 and M2, f is a strongly isotone map, and M3 says that f : XG1 −→ XG2 is continuous.
3.6
General I3 -frames and their morphisms
Definition 3.7. (General I3 -frame) Let G = hX, ≤, R, Ai be a general frame. G is a general
I3 -frame iff the following list of conditions is satisfied:
D1. XG is an Esakia space, and A is the collection of the clopen up-sets of XG .
D2. A is closed under 3R .
D3. For every x ∈ X, R[x] ∈ K(XG ).
Definition 3.8. (p-morphism of general I3 -frames) Let Gi = hXi , ≤, Ri , Ai i be general I3 frames, i = 1, 2. A map f : X1 → X2 is a p-morphism iff the following list of conditions is
satisfied for every x, x0 , y ∈ X1 , z ∈ X2 :
M1. if x ≤1 y then f (x) ≤2 f (y).
M2. If f (x) ≤2 z then f (x0 ) = z for some x0 ∈ x↑.
M3. For every Y ∈ A2 , f −1 [Y ] ∈ A1 .
M4. If xR1 y then f (x)R2 f (y).
M5. If f (x)R2 z then z ≤2 f (x0 ) for some x0 ∈ R1 [x].
3.9
General IK-frames and their morphisms
Definition 3.10. (General IK-frame) Let G = hX, ≤, R, Ai be a general frame. G is a general
IK-frame iff the following list of conditions is satisfied:
D1. XG is an Esakia space, and A is the collection of the clopen up-sets of XG .
D2. A is closed under 3R and 2(≤◦R) .
D3. For every x ∈ X, R[x] ∈ K(XG ).
D4. For every x ∈ X, R[x↑] ∈ K ↑ (XG ).
Example 3.11. For every finite partial order hX, ≤i, the general frame G = hX, ≤, (≥◦≤), Up(X)i
is a general IK-frame.
Proof. By definition, the topology τ of XG = X is generated by taking Up(X) ∪ Down(X) as a
subbase. As X is finite, then X = hX, ≤, τ i is compact and τ is the discrete topology, i.e. every
set is clopen. So it trivially follows that X is totally order-disconnected, hence it is a Priestley
space, and that for every clopen subset U of X, U ↓ is clopen, so X is an Esakia space. 2.2 (2)
implies that Up(X) is closed under 3(≥◦≤) , and by 2.3, Up(X) is closed under 2≤◦(≥◦≤) . For
every x ∈ X, (≥ ◦ ≤)[x] = x↓↑ ∈ Up(X) and (≥ ◦ ≤)[x↑] = x↑↓↑ ∈ Up(X), so they are clopen
up-sets, therefore (≥ ◦ ≤)[x] ∈ K(X) and (≥ ◦ ≤)[x↑] ∈ K ↑ (X).
Definition 3.12. (p-morphism of general IK-frames) Let Gi = hXi , ≤, Ri , Ai i be general
IK-frames, i = 1, 2. A map f : X1 → X2 is a p-morphism iff the following list of conditions is
satisfied for every x, x0 , y ∈ X1 , z ∈ X2 :
M1. if x ≤ y then f (x) ≤ f (y).
M2. If f (x) ≤ z then f (x0 ) = z for some x0 ∈ x↑.
M3. For every Y ∈ A2 , f −1 [Y ] ∈ A1 .
M4. If xR1 y then f (x)R2 f (y).
155
M5. If f (x)R2 z then z ≤ f (x0 ) for some x0 ∈ R1 [x].
M6. If f (x)(≤ ◦ R2 )z then f (x0 ) ≤ z for some x0 ∈ R1 [x↑].
4
From general frames to algebras
For every general frame G = hX, ≤, R, Ai, let G + := A, and for every continuous f : XG1 −→ XG2
let f + : G2+ −→ G1+ be given by the assignment Y 7−→ f −1 [Y ] for every Y ∈ AG2 .
In this section, we are showing that these assignments define functors ( )+ : LGF −→ LAlg for
L ∈ {I2 , I3 , IK}.
4.1
The action of ( )+ on objects
Let us recall that for every general frame G = hX, ≤, R, Ai, FG = hX, ≤, Ri is the associated
frame.
Lemma 4.2. Let G = hX, ≤, R, Ai be a general frame.
1. If G is a general I2 -frame, then FG is an I2 -frame.
2. If G is a general I3 -frame, then FG is an I3 -frame.
3. If G is a general IK-frame, then FG is an IK-frame.
Proof. 1. Let us show that for every x ∈ X, (≤ ◦ R)[x] ⊆ (R ◦ ≤)[x]: Suppose that z ∈ (≤ ◦ R)[x]
and z ∈
/ (R ◦ ≤)[x] = R[x]↑ for some z ∈ X. As z ∈
/ R[x]↑, then y 6≤ z for every y ∈ R[x], hence,
by D1, forS
every y ∈ R[x] there exists a clopen up-set Uy of XG such that y ∈ Uy and
/ Uy , and
Szn ∈
so R[x] ⊆ y∈R[x] Uy , and as XG is compact and R[x] is closed by D3, then R[x] ⊆ i=1 Uyi = U
for some y1 , . . . , yn ∈ R[x]. As U is a clopen up-set, then U ∈ A, moreover, z ∈
/ U and R[x] ⊆ U .
As z ∈ (≤ ◦ R)[x], then x ≤ wRz for some w ∈ X. Since z ∈ (R[w] \ U ), then w ∈
/ 2R U ∈ A by
D2’, so in particular 2R U is an up-set, and as x ≤ w, then x ∈
/ 2R U , i.e. R[x] 6⊆ U , contradiction.
2. Let us show that for every x ∈ X, (≥ ◦ R)[x] ⊆ (R ◦ ≥)[x]: Suppose that z ∈ (≥ ◦ R)[x] and
z∈
/ (R ◦ ≥)[x] = R[x]↓ for some z ∈ X. As z ∈
/ R[x]↓, then z 6≤ y for every y ∈ R[x], hence, by
D1, for every
y
∈
R[x]
there
exists
a
clopen
down-set
Vy of XG such that y ∈ Vy andSz ∈
/ Vy , and
S
n
so R[x] ⊆ y∈R[x] Vy , and as XG is compact and R[x] is closed by D3, then R[x] ⊆ i=1 Vyi = V
for some y1 , . . . , yn ∈ R[x]. Let U = (X \ V ). As U is a clopen up-set, then U ∈ A, moreover,
z ∈ U and R[x] ∩ U = ∅.
As z ∈ (≥ ◦ R)[x], then x ≥ wRz for some w ∈ X. Since z ∈ R[w] ∩ U , then w ∈ 3R U ∈ A by D2,
so in particular 3R U is an up-set, and as w ≤ x, then x ∈ 3R U , i.e. R[x] ∩ U 6= ∅, contradiction.
3. Let us show that for every x ∈ X, (R ◦ ≤)[x] ⊆ (≤ ◦ R)[x]: Suppose that z ∈ (R ◦ ≤)[x] and
z∈
/ (≤ ◦ R)[x] = R[x↑] for some z ∈ X. As z ∈
/ R[x↑] which is a closed up-set of XG by D4, then
y 6≤ z for every y ∈ R[x↑], hence, by D1, for everySy ∈ R[x↑] there exists a clopen up-set Uy of
XG such that y ∈ Uy and z ∈
/ U , and so R[x↑] ⊆ y∈R[x] Uy , and as XG is compact and R[x] is
Sn y
closed by D3, then R[x] ⊆ i=1 Uyi = U for some y1 , . . . , yn ∈ R[x]. As U is a clopen up-set, then
U ∈ A, moreover, z ∈
/ U and R[x↑] ⊆ U .
As z ∈ (R ◦ ≤)[x], then xRw ≤ z for some w ∈ X. Since w ∈ R[x] ⊆ R[x↑] ⊆ U , then w ∈ U
which is an up-set, and as w ≤ z, then z ∈ U , contradiction. The proof of the other condition is
the same as the proof of (2).
Proposition 4.3. Let L ∈ {I2 , I3 , IK}. For every general L-frame G = hX, ≤, R, Ai, A is an
L-algebra.
Proof. It immediately follows from 2.4 and 4.2.
156
4.4
The action of ( )+ on arrows
Proposition 4.5. Let L ∈ {I2 , I3 , IK}. For every p-morphism h : G1 −→ G2 of general L-frames,
h+ : G2 + −→ G1 + is a homomorphism of L-algebras.
Proof. If h : G1 −→ G2 is a p-morphism of general L-frames, then in particular it is a continuous
and strongly isotone map between the Esakia spaces XG1 and XG2 , hence from the duality for
Heyting algebras, h+ is a homomorphism between the Heyting algebra reducts of G2 + and G1 + .
Let us show that if G1 and G2 are general I2 -frames, then for every Y ∈ A2 ,
h−1 [2R2 Y ] = 2R1 h−1 [Y ].
For every x ∈ X1 , x ∈ h−1 [2R2 Y ] iff R2 [h(x)] ⊆ Y , and x ∈ 2R1 h−1 [Y ] iff R1 [x] ⊆ h−1 [Y ].
(⊆) Assume that z ∈ R1 [x] and show that z ∈ h−1 [Y ]: As xR1 z, then, by M4, h(x)R2 h(z), i.e.
h(z) ∈ R2 [h(x)] ⊆ Y , hence z ∈ h−1 [Y ].
(⊇) Assume that z ∈ R2 [h(x)] and show that z ∈ Y : If h(x)R2 z, then, by M5’, there exists
y ∈ R1 [x] ⊆ h−1 [Y ] such that h(y) ≤ z. As h(y) ∈ Y and Y is an up-set, then z ∈ Y .
Let us show that if G1 and G2 are general I3 -frames, then for every Y ∈ A2 ,
h−1 [3R2 Y ] = 3R1 h−1 [Y ].
For every x ∈ X1 , x ∈ h−1 [3R2 Y ] iff R2 [h(x)] ∩ Y 6= ∅, and x ∈ 3R1 h−1 [Y ] iff R1 [x] ∩ h−1 [Y ] 6= ∅.
(⊆) Assume that z ∈ R2 [h(x)] ∩ Y . As h(x)R2 z, then, by M5, there exists y ∈ R1 [x] such that
z ≤2 h(y). As z ∈ Y and Y is an up-set, then h(y) ∈ Y . Hence y ∈ R1 [x] ∩ h−1 [Y ] 6= ∅.
(⊇) Assume that z ∈ R1 [x] ∩ h−1 [Y ], hence h(z) ∈ Y and xR1 z, so, by M4, h(x)R2 h(z), i.e.
h(z) ∈ R2 [h(x)], and so h(z) ∈ R2 [h(x)] ∩ Y 6= ∅.
Let us show that if G1 and G2 are general IK-frames, then for every Y ∈ A2 ,
h−1 [2(≤◦R2 ) Y ] = 2(≤◦R1 ) h−1 [Y ].
For every x ∈ X1 , x ∈ h−1 [2(≤◦R2 ) Y ] iff (≤ ◦ R2 )[h(x)] ⊆ Y , and x ∈ 2(≤◦R1 ) h−1 [Y ] iff
(≤ ◦ R1 )[x] ⊆ h−1 [Y ].
(⊆) Assume that z ∈ (≤ ◦ R1 )[x] and show that z ∈ h−1 [Y ]: As x ≤ wR1 z for some w ∈ X1 ,
then, by M1 and M4, h(x) ≤ h(w)R2 h(z), i.e. h(z) ∈ (≤ ◦ R2 )[h(x)] ⊆ Y , hence z ∈ h−1 [Y ].
(⊇) Assume that z ∈ (≤ ◦ R2 )[h(x)] and show that z ∈ Y : If h(x)(≤ ◦ R2 )z, then, by M5, there
exists y ∈ (≤ ◦ R1 )[x] ⊆ h−1 [Y ] such that h(y) ≤ z. As h(y) ∈ Y and Y is an up-set, then z ∈ Y .
The proof that h−1 [3R2 Y ] = 3R1 h−1 [Y ] goes as in the I3 -case.
5
From algebras to general frames
Let L ∈ {I2 , I3 , IK}. For every L-algebra A let P r(A) be the collection of the prime filters of the
L-reduct of A. Let us define A+ := hP r(A), ⊆, RA , Ai, where for every P, Q ∈ P r(A):
R1. If A is an I2 -algebra, P RA Q iff 2−1 [P ] ⊆ Q.
R2. If A is an I3 -algebra, P RA Q iff Q ⊆ 3−1 [P ].
R3. If A is an IK-algebra, P RA Q iff 2−1 [P ] ⊆ Q ⊆ 3−1 [P ].
As for the definition of A, for every a ∈ A, let a = {P ∈ P r(A) | a ∈ P }. Then the carrier of A is
A = {a | a ∈ A}. For every n-ary operation ∗ in the signature of A, ∗A (a1 , . . . , an ) = ∗(a1 , . . . , an ).
The operations ∗A are well-defined, for if ai = ci , then for every P ∈ P r(A), ai ∈ P iff ci ∈ P ,
and by Birkhoff-Stone theorem, this implies that ai = ci . From the Priestley duality restricted
to Heyting algebras it follows that the L-reduct of A is a subalgebra of the Heyting algebra
hUp(P r(A)), ∩, ∪, ⇒, ∅, P r(A)i.
For every homomorphism f : A1 −→ A2 let f+ : A2 + −→ A1 + be given by the assignment
P 7−→ f −1 [P ] for every P ∈ P r(A2 ).
In this section, we are showing that these assignments define functors ( )+ : LAlg −→ LGF for
L ∈ {I2 , I3 , IK}.
157
5.1
Properties of RA
Lemma 5.2. For every L-algebra A, RA is a closed subset of XA+ × XA+ .
Proof. Assume that RA is defined like in R1. If hP, Qi ∈
/ RA , then 2−1 [P ] 6⊆ Q, i.e. 2a ∈ P and
a∈
/ Q for some a ∈ A. Hence P ∈ (2a) and Q ∈
/ a. Let us consider U = (2a) × (P r(A) \ a). U is
an open subset of XA+ × XA+ , for both (2a) and P r(A) \ a are open subsets of XA+ , moreover
hP, Qi ∈ U. Let us show that RA ∩ U = ∅: If hS, T i ∈ U , then 2a ∈ S and a ∈
/ T , hence
2−1 [S] 6⊆ T , i.e. hS, T i ∈
/ RA .
Assume that RA is defined like in R2. If hP, Qi ∈
/ RA , then Q 6⊆ 3−1 [P ], i.e. a ∈ Q and 3a ∈
/P
for some a ∈ A. Hence Q ∈ a and P ∈
/ (3a). Let us consider U = (P r(A) \ (3a)) × a. U is an open
subset of XA+ ×XA+ , for both P r(A)\(3a) and a are open subsets of XA+ , moreover hP, Qi ∈ U.
Let us show that RA ∩ U = ∅: If hS, T i ∈ U, then 3a ∈
/ S and a ∈ T , hence T 6⊆ 3−1 [S], i.e.
hS, T i ∈
/ RA .
Assume that RA is defined like in R3. Then RA is the intersection of two sets that, by the
previous cases, are closed, and so RA is closed.
Corollary 5.3. For every L-algebra A, if X is a closed subset of XA+ , then RA [X ] is a closed
subset of XA+ .
Proof. For i = 1, 2 let πi : XA+ × XA+ −→ XA+ be the canonical projections. For every closed
subset X of XA+ ,
RA [X ]
=
=
{Q ∈ P r(A) | P RA Q for some P ∈ X }
π2 [RA ∩ (X × P r(A))].
By 5.2 RA is closed, hence so is RA ∩ (X × P r(A)), and as π2 is a closed map, for it is a continuous
map between compact spaces, then π2 [RA ∩ (X × P r(A))] is closed.
Lemma 5.4.
1. For every I2 -algebra A, RA = (⊆ ◦ RA ◦ ⊆).
2. For every I3 -algebra A, RA = (⊇ ◦ RA ◦ ⊇).
3. For every IK-algebra A, RA = (⊆ ◦ RA ) ∩ (RA ◦ ⊇).
Proof. We prove only the inclusions from right to left, being the converse inclusions immediate.
1. If hP, Qi ∈ (⊆ ◦ RA ◦ ⊆), then P ⊆ S1 RA S2 ⊆ Q, for some S1 , S2 ∈ P r(A), hence 2−1 [P ] ⊆
2−1 [S1 ] ⊆ S2 ⊆ Q, i.e. hP, Qi ∈ RA .
2. If hP, Qi ∈ (⊇ ◦ RA ◦ ⊇), then P ⊇ S1 RA S2 ⊇ Q, for some S1 , S2 ∈ P r(A), hence Q ⊆ S2 ⊆
3−1 [S1 ] ⊆ 3−1 [P ], i.e. hP, Qi ∈ RA .
3. If hP, Qi ∈ (⊆ ◦ RA ) ∩ (RA ◦ ⊇), then P ⊆ S1 RA Q and P RA S2 ⊇ Q for some S1 , S2 ∈ P r(A),
then 2−1 [P ] ⊆ 2−1 [S1 ] ⊆ Q and Q ⊆ S2 ⊆ 3−1 [P ], hence P RA Q.
For every LM -algebra A and every subset B of A, let F i(B) (Id(B)) be the filter (ideal) of the
lattice reduct of A that is generated by B. B c is the complement of B in A.
Lemma 5.5. For every IK-algebra A and every P, Q ∈ P r(A),
1. hP, Qi ∈ (RA ◦ ⊇) iff Q ⊆ 3−1 [P ].
2. hP, Qi ∈ (⊆ ◦ RA ) iff 2−1 [P ] ⊆ Q.
Proof. 1. ‘if’: Assume that Q ⊆ 3−1 [P ], and let us show that there exists S ∈ P r(A) such that
P RA S ⊇ Q, i.e. such that Q ∪ 2−1 [P ] ⊆ S and S ∩ 3−1 [P ]c = ∅. Let us consider F i(Q ∪ 2−1 [P ]):
If we show that
F i(Q ∪ 2−1 [P ]) ∩ 3−1 [P ]c = ∅,
158
then the statement will follow by Birkhoff-Stone theorem. Suppose that F i(Q ∪ 2−1 [P ]) ∩
3−1 [P ]c 6= ∅. Then there exists c ∈ A such that 3c ∈
/ P and a ∧ b ≤ c for some a ∈ 2−1 [P ] and
b ∈ Q. Then b ≤ a → c, hence 3b ≤ 3(a → c) ≤ (2a → 3c). As b ∈ Q ⊆ 3−1 [P ], then 3b ∈ P ,
hence 2a → 3c ∈ P , and as 2a ∈ P , then 3c ∈ P , contradiction.
‘only if’: If P RA S ⊇ Q for some S ∈ P r(A), then Q ⊆ S ⊆ 3−1 [P ].
2. ‘if’: Assume that 2−1 [P ] ⊆ Q, and let us show that there exists S ∈ P r(A) such that P ⊆ S
and 2−1 [P ] ⊆ Q ⊆ 3−1 [P ], i.e. such that P ∪ 3[P ] ⊆ S and S ∩ 2[Qc ] = ∅. Let us consider
F i(P ∪ 3[Q]): If we show that
F i(P ∪ 3[Q]) ∩ 2[Qc ] = ∅,
then the statement will follow by Birkhoff-Stone theorem. Suppose that F i(P ∪ 3[Q]) ∩ 2[Qc ] 6= ∅.
Then there exist a ∈ Qc , b ∈ P and c ∈ Q such that b ∧ 3c ≤ 2a. Then b ≤ 3c → 2a ≤ 2(c → a).
As b ∈ P , then 2(c → a) ∈ P , hence c → a ∈ 2−1 [P ] ⊆ Q, and as c ∈ Q, then a ∈ Q,
contradiction.
‘only if’: If P ⊆ SRA Q for some S ∈ P r(A), then 2−1 [P ] ⊆ 2−1 [S] ⊆ Q.
Corollary 5.6.
1. For every I2 -algebra A and every P ∈ P r(A), if 2a ∈
/ P , then a ∈
/ Q and P RA Q for some
Q ∈ P r(A).
2. For every I3 -algebra A and every P ∈ P r(A), if 3a ∈ P , then a ∈ Q and P RA Q for some
Q ∈ P r(A).
/ P , then a ∈
/ Q and P ⊆ SRA Q for some Q, S ∈ P r(A).
3. For every IK-algebra A, if 2a ∈
4. For every IK-algebra A and every P ∈ P r(A), if 3a ∈ P , then a ∈ S and P RA S for some
S ∈ P r(A).
/ P , then Id(a) ∩ 2−1 [P ] = ∅, for if not, then c ≤ a for some c such that 2c ∈ P ,
Proof. 1. If 2a ∈
hence 2c ≤ 2a, and so 2a ∈ P , contradiction. By Birkhoff-Stone theorem, there exists Q ∈ P r(A)
such that 2−1 [P ] ⊆ Q, i.e. P RA Q, and a ∈
/ Q.
2. If 3a ∈ P , then F i(a) ∩ 3−1 [P c ] = ∅, for if not, then a ≤ c for some c such that 3c ∈
/ P , hence
3a ≤ 3c, and so 3c ∈ P , contradiction. By Birkhoff-Stone theorem, there exists Q ∈ P r(A) such
that a ∈ Q and Q ⊆ 3−1 [P ], i.e. P RA Q.
3. If 2a ∈
/ P , then Id(a) ∩ 2−1 [P ] = ∅, so by Birkhoff-Stone theorem, there exists Q ∈ P r(A)
such that 2−1 [P ] ⊆ Q and a ∈
/ Q. By 5.5 (2), P ⊆ SRA Q for some S ∈ P r(A).
4. If 3a ∈ P , then F i(a) ∩ 3−1 [P c ] = ∅, so by Birkhoff-Stone theorem, there exists Q ∈ P r(A)
such that a ∈ Q and Q ⊆ 3−1 [P ]. By 5.5 (1), P RA S ⊇ Q for some S ∈ P r(A), and as a ∈ S ⊆ Q,
then a ∈ S.
Corollary 5.7.
1. For every I2 -algebra A, (⊆ ◦ RA ) ⊆ (RA ◦ ⊆).
2. For every I3 -algebra A, (⊇ ◦ RA ) ⊆ (RA ◦ ⊇).
3. For every IK-algebra A, (⊇ ◦ RA ) ⊆ (RA ◦ ⊇) and (RA ◦ ⊆) ⊆ (⊆ ◦ RA ).
Proof. 1. If P ⊆ SRA Q, then 2−1 [P ] ⊆ 2−1 [S] ⊆ Q, hence P RA Q ⊆ Q.
2. If P ⊇ SRA Q, then Q ⊆ 3−1 [S] ⊆ 3−1 [P ], hence P RA Q ⊇ Q.
3. If P RA S ⊆ Q, then 2−1 [P ] ⊆ S ⊆ Q, hence by 5.5 (2), hP, Qi ∈ (⊆ ◦ RA ).
If P ⊇ SRA Q, then Q ⊆ 3−1 [S] ⊆ 3−1 [P ], hence by 5.5 (1), hP, Qi ∈ (RA ◦ ⊇).
159
5.8
The action of ( )+ on objects
Proposition 5.9. Let L ∈ {I2 , I3 , IK}. For every L-algebra A, A+ = hP r(A), ⊆, RA , Ai is a
general L-frame.
Proof. The Priestley duality restricted to Heyting algebras yields that XA+ is an Esakia space,
and A is the collection of the clopen up-sets of XA+ , which is D1. As XA+ is an Esakia space,
then in particular it is Hausdorff, hence for every P ∈ P r(A), {P } is closed in XA+ , and so by
5.3, RA [P ] is closed in XA+ , which is D3.
Let us show that if A is an I2 -algebra, then 2A = 2RA , i.e. that for every a ∈ A,
(2a) = 2RA a.
(⊆) If P ∈ (2a), then 2a ∈ P , i.e. a ∈ 2−1 [P ] so, for every Q ∈ P r(A), if P RA Q, then
a ∈ 2−1 [P ] ⊆ Q.
(⊇) If P ∈
/ (2a), then by item 1 of 5.6, a ∈
/ Q and P RA Q for some Q ∈ P r(A), so P ∈
/ 2RA a.
Let us show that if A is an I3 -algebra (an IK-algebra), then 3A = 3RA , i.e. that for every a ∈ A,
(3a) = 3RA a.
(⊆) If P ∈ (3a), then 3a ∈ P , then by item 2 (item 4) of 5.6, a ∈ Q and P RA Q for some
Q ∈ P r(A), hence P ∈ 3RA a.
(⊇) If P ∈ 3RA a, then a ∈ Q and P RA Q for some Q ∈ P r(A), i.e. Q ⊆ 3−1 [P ], hence 3a ∈ P .
Let us show that if A is an IK-algebra, then 2A = 2(⊆◦RA ) , i.e. that for every a ∈ A,
(2a) = 2(⊆◦RA ) a.
(⊆) If P ∈ (2a), then 2a ∈ P , i.e. a ∈ 2−1 [P ] so, for every Q ∈ P r(A), if P RA Q, then
a ∈ 2−1 [P ] ⊆ Q.
(⊇) If P ∈
/ (2a), then by item 3 of 5.6, a ∈
/ Q and P ⊆ SRA Q for some Q, S ∈ P r(A), so
P ∈
/ 2(⊆◦RA ) a.
This is enough for proving that A is closed in each case under the appropriate operations.
If A is an IK-algebra, then as XA+ is an Esakia space, then in particular it is a Priestley space,
so by 3.2 for every P ∈ P r(A), P ↑ = {Q ∈ P r(A) | P ⊆ Q} is a closed subset of XA+ , hence by
5.3, RA [P ↑] is closed in XA+ .
Let us show that RA [P ↑] is an up-set: If Q ∈ RA [P ↑] and Q ⊆ T , then P ⊆ SRA Q ⊆ T , hence
by 5.7 (3), P ⊆ S ⊆ Q0 RA T , and so T ∈ RA [P ↑]. This proves D4.
5.10
The action of ( )+ on arrows
Proposition 5.11. Let L ∈ {I2 , I3 , IK}. For every L-algebra homomorphism h : A1 −→ A2 ,
h+ : A2 + −→ A1 + is a p-morphism of general L-frames.
Proof. The Priestley duality restricted to Heyting algebras yields that h+ is a continuous and
strongly isotone map between XA2 + and XA1 + , which is equivalent to conditions M1–M3.
Let us show that if P, Q ∈ P r(A2 ) and 2−1 [P ] ⊆ Q, then 2−1 [h−1 [P ]] ⊆ h−1 [Q]: For every
a ∈ A2 ,
a ∈ 2−1 [h−1 [P ]]
⇔
⇔
⇔
⇔
⇒
2a ∈ h−1 [P ]
h(2a) ∈ P
2h(a) ∈ P
h(a) ∈ 2−1 [P ] ⊆ Q
a ∈ h−1 [Q].
Let us show that if P, Q ∈ P r(A2 ) and Q ⊆ 3−1 [P ], then h−1 [Q] ⊆ 3−1 [h−1 [P ]]: For every
a ∈ A2 ,
160
a ∈ h−1 [Q]
⇔
⇒
⇔
⇔
⇔
h(a) ∈ Q ⊆ 3−1 [P ]
3h(a) ∈ P
h(3a) ∈ P
3a ∈ h−1 [P ]
a ∈ 3−1 [h−1 [P ]].
This is enough for proving that for L ∈ {I2 , I3 , IK} and for every P, Q ∈ P r(A2 ), if P RA2 Q,
then h+ (P )RA1 h+ (Q), which is M4.
Let us show M5 for I3 -algebras, i.e. that if A1 and A2 are I3 -algebras and P ∈ P r(A2 ), Q ∈
P r(A1 ) are such that h−1 [P ]RA1 Q, then there exists S ∈ RA2 [P ] such that Q ⊆ h−1 [S]. We need
to find a prime filter S of A2 such that S ⊆ 3−1 [P ], i.e. S ∩ 3−1 [P ]c = ∅ and Q ⊆ h−1 [S], i.e.
h[Q] ⊆ S. It holds that
F i(h[Q]) ∩ 3−1 [P ]c = ∅,
for if not, then there are a ∈ Q and 3b ∈
/ P such that h(a) ≤ b, hence 3h(a) ≤ 3b. As
a ∈ Q ⊆ 3−1 [h−1 [P ]], then 3h(a) ∈ P , and so 3b ∈ P , contradiction.
By Birkhoff-Stone theorem, there exists S ∈ P r(A2 ) such that h[Q] ⊆ S (i.e. Q ⊆ h−1 [S]) and
S ∩ 3−1 [P ]c = ∅, i.e. S ⊆ 3−1 [P ], i.e. P RA2 S.
Let us show M5 for IK-algebras: Like before, it holds that F i(h[Q]) ∩ 3−1 [P ]c = ∅, so by
Birkhoff-Stone theorem, h[Q] ⊆ T (i.e. Q ⊆ h−1 [T ]) and T ∩ 3−1 [P ]c = ∅ for some T ∈ P r(A2 ).
As T ⊆ 3−1 [P ], then by 5.5 (1), hP, T i ∈ (RA2 ◦ ⊇), i.e. P RA2 S ⊇ T for some S ∈ P r(A2 ), so
S ∈ RA2 [P ] and Q ⊆ h−1 [T ] ⊆ h−1 [S].
Let us show M5’, i.e. that if A1 and A2 are I2 -algebras and P ∈ P r(A2 ), Q ∈ P r(A1 ) are such
that h−1 [P ]RA1 Q, then there exists S ∈ RA2 [P ] such that h−1 [S] ⊆ Q: what we need to hold is
that 2−1 [P ] ⊆ S and S ⊆ h[Q], i.e. S ∩ h[Qc ] = ∅. If we show that
h[Qc ] ∩ F i(2−1 [P ]) = ∅,
then the statement will follow from Birkhoff-Stone theorem. Suppose that there are a ∈
/ Q and
2b ∈ P such that b ≤ h(a), hence 2b ≤ 2h(a) = h(2a). As 2b ∈ P , then h(2a) ∈ P , hence
a ∈ 2−1 [h−1 [P ] ⊆ Q, contradiction.
Let us show M6, i.e. that if A1 and A2 are IK-algebras and P ∈ P r(A2 ), Q ∈ P r(A1 ) are such
that h−1 [P ](⊆ ◦ RA1 )Q, then there exists S ∈ (⊆ ◦ RA1 )[P ] such that h−1 [S] ⊆ Q: By 5.5 (2), we
need that 2−1 [P ] ⊆ S, moreover, we need that S ⊆ h[Q], i.e. S ∩ h[Qc ] = ∅. The proof goes like
in the case treated before.
6
Duality
In this section we are going to introduce the full subcategories LSp of LGF for L ∈ {I2 , I3 , IK},
and show that the functors ( )+ : LAlg −→ LSp and ( )+ : LSp −→ LAlg establish a duality.
For every general L-frame G = hX, ≤, R, Ai, let us consider the assignment εG : X −→ P(A)
that maps every x ∈ X to the set εG (x) = {Y ∈ A | x ∈ Y }. The Priestley duality restricted to
Heyting algebras yields that this assignment defines a map εG : XG −→ X(G + )+ that is an iso in
the category Es of Esakia spaces and continuous and strongly isotone maps (which is the dual of
the category of Heyting algebras and their homomorphisms).
So we will only need to show that the objects in LSp are exactly those general L-frames G such
that εG is an iso in LGF.
6.1
L-spaces
Definition 6.2. (I2 -space) Let G = hX, ≤, R, Ai be a general frame. G is an I2 -space iff the
following list of conditions is satisfied:
D1. XG is an Esakia space, and A is the collection of the clopen up-sets of XG .
D2’. A is closed under 2R .
D3. For every x ∈ X, R[x] ∈ K ↑ (XG ) = {F ∈ K(XG ) | F = F ↑}.
161
So I2 -spaces are those general I2 -frames such that R[x] is an up-set for every x ∈ X. Let I2 Sp
be the category of the I2 -spaces and their p-morphisms.
Example 6.3. For every finite partial order hX, ≤i, the general frame G = hX, ≤, ≤, Up(X)i is
an I2 -space.
Proof. By definition, the topology τ of XG = X is generated by taking Up(X) ∪ Down(X) as a
subbase. In 3.11, we saw that X is an Esakia space and that Up(X) is the collection of the clopen
up-sets of X. 2.2 (1) implies that Up(X) is closed under 2≤ . For every x ∈ X, x↑ ∈ Up(X) is a
clopen up-set, so in particular x↑ ∈ K ↑ (X).
Definition 6.4. (I3 -space) Let G = hX, ≤, R, Ai be a general frame. G is an I3 -space iff the
following list of conditions is satisfied:
D1. XG is an Esakia space, and A is the collection of the clopen up-sets of XG .
D2. A is closed under 3R .
D3. For every x ∈ X, R[x] ∈ K ↓ (XG ) = {F ∈ K(XG ) | F = F ↓}.
So I3 -spaces are those general I3 -frames such that R[x] is a down-set for every x ∈ X. Let I3 Sp
be the category of the I3 -spaces and their p-morphisms.
Example 6.5. For every finite partial order hX, ≤i, the general frame G = hX, ≤, ≥, Up(X)i is
an I3 -space.
Proof. By definition, the topology τ of XG = X is generated by taking Up(X) ∪ Down(X) as a
subbase. In 3.11, we saw that X is an Esakia space and that Up(X) is the collection of the clopen
up-sets of X. 2.2 (2) implies that Up(X) is closed under 3≥ . For every x ∈ X, x↓ ∈ Down(X) is
a clopen down-set, so in particular x↓ ∈ K ↓ (X).
Definition 6.6. (IK-space) Let G = hX, ≤, R, Ai be a general frame. G is an IK-space iff the
following list of conditions is satisfied:
D1. XG is an Esakia space, and A is the collection of the clopen up-sets of XG .
D2. A is closed under 3R and 2(≤◦R) .
D3. For every x ∈ X, R[x] ∈ K(XG ).
D4. For every x ∈ X, R[x↑] ∈ K ↑ (XG ).
D5. For every x ∈ X, R[x] = R[x↑] ∩ R[x]↓.
Let IKSp be the category of the IK-spaces and their p-morphisms. Conditions D4 and D5 together
imply that for every x ∈ X, R[x] is the intersection of an up-set and a down-set, hence R[x] is
convex, i.e. R[x] = R[x]↑ ∩ R[x]↓1 . So if G is an IK-space, then G is a general IK-frame such
that R[x] is convex for every x ∈ X. However, not all the general IK-frames such that R[x] is
convex for every x ∈ X are IK-spaces: Indeed, we saw already that, given a finite partial order
hX, ≤i, the general frame G = hX, ≤, (≥ ◦ ≤), Up(X)i is general IK-frame, and moreover for every
x ∈ X, (≥ ◦ ≤)[x] = x↓↑ is an up-set, hence it is convex. However, G is not an IK-space in general.
Consider the partial order associated with the following Hasse diagram:
zu
zu
1
4
¡@
¡@
¡
¡
@
@ y
x¡
u
@u
@¡
¡@
@
¡
¡
¡
@
@
@¡
@¡
u
u
z2
z3
The relation (≥ ◦ ≤) does not satisfy D5: It holds that x ≤ z1 ≥ z2 ≤ y, so y ∈ (≥ ◦ ≤)[x↑], and
x ≥ z3 ≤ z4 ≥ y, so y ∈ (≥ ◦ ≤)[x]↓, but y ∈
/ (≥ ◦ ≤)[x].
1 It is very easy to show that for every partial order hX, ≤i and every Y ⊆ X, Y = Y ↑ ∩ Y ↓ (i.e. Y is convex) iff
Y = U ∩ V for some U ∈ Up(X) and V ∈ Down(X).
162
Example 6.7. For every finite linear order hX, ≤i, the general IK-frame G = hX, ≤, (≥◦≤), Up(X)i
is an IK-space.
Proof. Since ≤ is a linear order, then for every x ∈ X, X = x↑ ∪ x↓ ⊆ (≥ ◦ ≤)[x], hence
(≥ ◦ ≤)[x] = (≥ ◦ ≤)[x↑] ∩ (≥ ◦ ≤)[x]↓, which is D5.
Proposition 6.8. For every L-algebra A, A+ is an L-space.
Proof. By 5.9, A+ is a general L-frame. By 5.4 (1), if A is an I2 -algebra, then RA [P ] = (⊆ ◦ RA ◦
⊆)[P ] is an up-set for every P ∈ P r(A). Analogously, 5.4 (2) and (3) respectively imply that if A
is an I3 -algebra, then RA [P ] is a down-set for every P ∈ P r(A), and if A is an IK-algebra, then
RA [P ] = (⊆ ◦ RA )[P ] ∩ (RA ◦ ⊇)[P ] for every P ∈ P r(A).
Lemma 6.9. For every general L-frame G = hX, ≤, R, Ai and every x, y ∈ X,
1. x ≤ y iff εG (x) ⊆ εG (y).
2. If xRy then εG (x)RA εG (y).
Proof. 1. If x ≤ y then, as A ⊆ Up(X), for every Y ∈ A, if x ∈ Y then y ∈ Y .
If x 6≤ y then, as XG is totally order-disconnected and A is the collection of the clopen up-sets of
XG , x ∈ Y and y ∈
/ Y for some Y ∈ A, hence Y ∈ (εG (x) \ εG (y)), and so εG (x) 6⊆ εG (y).
−1
2. Let us show that if y ∈ R[x], then a) 2−1
R [εG (x)] ⊆ εG (y), b) εG (y) ⊆ 3R [εG (x)] and c)
−1
2(≤◦R) [εG (x)] ⊆ εG (y):
a) For every Y ∈ A, 2R Y ∈ εG (x) iff x ∈ 2R Y , iff R[x] ⊆ Y , and so y ∈ Y , i.e. Y ∈ εG (y).
b) For every Y ∈ A, Y ∈ εG (y) iff y ∈ Y , and as y ∈ R[x], then R[x] ∩ Y 6= ∅, i.e. 3R Y ∈ εG (x),
i.e. Y ∈ 3−1
R [εG (x)].
c) For every Y ∈ A, 2(≤◦R) Y ∈ εG (x) iff x ∈ 2(≤◦R) Y , iff (≤ ◦ R)[x] ⊆ Y , and so y ∈ R[x] ⊆
(≤ ◦ R)[x] ⊆ Y , i.e. Y ∈ εG (y).
a) proves the statement if A is an I2 -algebra, b) proves the statement if A is an I3 -algebra, and
a) and c) together prove the statement if A is an IK-algebra.
The following lemma says that the additional conditions that L-spaces satisfy are exactly what
is needed in each situation in order for ε to be an iso.
Lemma 6.10.
1. The following are equivalent for every general I2 -frame:
(a) For every x ∈ X, R[x] = R[x]↑.
(b) For every x, y ∈ X, if ε(x)RA ε(y) then xRy.
2. The following are equivalent for every general I3 -frame:
(a) For every x ∈ X, R[x] = R[x]↓.
(b) For every x, y ∈ X, if ε(x)RA ε(y) then xRy.
3. The following are equivalent for every general IK-frame:
(a) For every x ∈ X, R[x] = R[x↑] ∩ R[x]↓.
(b) For every x, y ∈ X, if ε(x)RA ε(y) then xRy.
Proof. 1. (a ⇒ b) Suppose that x, y ∈ X are such that ε(x)RA ε(y) but y ∈
/ R[x] = R[x]↑. Then
R[x] ⊆ U and y ∈
/ U for some U ∈ A, hence x ∈ 2R U . As ε(x)RA ε(y), then 2−1
R [ε(x)] ⊆ ε(y), i.e.
for every U ∈ A, if x ∈ 2R U , then y ∈ U , contradiction.
(b ⇒ a) (⊇) If y ∈ R[x]↑, then xRz ≤ y for some z ∈ X, hence, by 6.9, ε(x)RA ε(z) ⊆ ε(y), i.e.
2−1
R [ε(x)] ⊆ ε(z) ⊆ ε(y), hence ε(x)RA ε(y), and so by assumption it follows that xRy.
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2. (a ⇒ b) Suppose that x, y ∈ X are such that ε(x)RA ε(y) but y ∈
/ R[x] = R[x]↓. Then
y ∈ U and R[x] ∩ U = ∅ for some clopen up-set subset U , hence x ∈
/ 3R U . As ε(x)RA ε(y), then
−1
ε(y) ⊆ 3R
[ε(x)], i.e. for every U ∈ A, if y ∈ U then x ∈ 3R U , contradiction.
(b ⇒ a) (⊇) If y ∈ R[x]↓, then xRz ≥ y for some z ∈ X, hence, by 6.9, ε(x)RA ε(z) ⊇ ε(y), i.e.
ε(y) ⊆ ε(z) ⊆ 3−1
R [ε(x)], hence ε(x)RA ε(y), and so by assumption it follows that xRy.
3. (a ⇒ b) Suppose that x, y ∈ X are such that ε(x)RA ε(y) but y ∈
/ R[x] = R[x↑] ∩ R[x]↓.
Then either y ∈
/ R[x↑] or y ∈
/ R[x]↓. If y ∈
/ R[x↑] = R[x↑]↑ Then R[x↑] ⊆ U and y ∈
/ U for some
U ∈ A, hence x ∈ 2(≤◦R) U . As ε(x)RA ε(y), then 2−1
[ε(x)]
⊆
ε(y),
i.e.
for
every
U ∈ A, if
(≤◦R)
x ∈ 2(≤◦R) U , then y ∈ U , contradiction. If y ∈
/ R[x]↓ the proof is analogous to the proof of (2),
(a ⇒ b).
(b ⇒ a) (⊇) If y ∈ R[x↑] ∩ R[x]↓, then x ≤ z1 Ry and xRz2 ≥ y for some z1 , z2 ∈ X, hence,
−1
by 6.9, ε(x) ⊆ ε(z1 )RA ε(y) and ε(x)RA ε(z2 ) ⊇ ε(y), and so 2−1
R [ε(x)] ⊆ 2R [ε(z1 )] ⊆ ε(y) and
−1
ε(y) ⊆ ε(z2 ) ⊆ 3R [ε(x)], hence ε(x)RA ε(y), and so by assumption it follows that xRy.
Proposition 6.11. For every L-space G = hX, ≤, R, Ai, εG : G −→ (G + )+ is a p-morphism of
L-spaces, hence it is an iso in LGF.
Proof. From the duality for Heyting algebras, we know that εG : XG −→ X(G + )+ is an iso in Es,
hence it is bijective and satisfies M1–M3. M4 holds by 6.9 (2). The surjectivity of εG and 6.10
imply M5’, M5 and M6. Let us show M6: If εG (x)(⊆ ◦ RA )P = εG (y), then εG (x) ⊆ εG (z)RA εG (y)
for some z ∈ X, hence, by item 1 of 6.9 and 6.10, x ≤ zRy, i.e. y ∈ (≤ ◦ R)[x].
Theorem 6.12. For every L ∈ {I2 , I3 , IK}, the category LAlg of L-algebras and their homomorphisms is dually equivalent to the category LSp of L-spaces and their p-morphisms.
Proof. It follows from 4.3, 4.5, 5.11, 6.8, and 6.11.
7
Characterizing topological semantics of MIPC
Bezhanishvili [1, 2] introduced a topological semantics for MIPC, given by the category TPSOE of
perfect augmented Kripke frames and their morphisms (see 7.6 and 7.10 below), and proved that
TPSOE is dually equivalent to the category of monadic Heyting algebras and their homomorphisms,
that is the class of algebras canonically associated with MIPC (see [2]). In this section, we will
show that – as was to be expected – TPSOE is isomorphic to the full subcategory MIPCSp of
IKSp whose objects are the MIPC-spaces defined below:
Definition 7.1. (MIPC-space) An MIPC-space is an IK-space G = hX, ≤, E, Ai such that E
is an equivalence relation.
Definition 7.2. (Augmented Kripke frame) (cf. Def 2.1 of [2]) A relational structure hX, ≤, Ei
is an augmented Kripke frame iff hX, ≤i is a partial order and E is an equivalence relation on X
such that (E ◦ ≤) ⊆ (≤ ◦ E).
Lemma 7.3. For every relational structure F = hX, ≤, Ei, F is an augmented Kripke frame iff
F is an IK-frame such that ≤ is a partial order and E is an equivalence relation.
Proof. The ‘if’ direction is immediate. As for the converse, if F is an augmented Kripke frame we
only need to show that (≥ ◦ E) ⊆ (E ◦ ≥): if x, y, z ∈ X and x ≥ yEz, then, as E is symmetric,
zEy ≤ x, and so z ≤ vEx for some v ∈ X, hence xEv ≥ z.
Definition 7.4. (Perfect Kripke frame) (cf. Section 3.1 of [2]) A preordered Stone space X =
hX, ≤, τ i is a perfect Kripke frame iff x↑ ∈ K(X) for every x ∈ X and for every clopen subset U
of X, U ↓ = ≤−1 [U ] is clopen.
Lemma 7.5. For every perfect Kripke frame X = hX, ≤, τ i,
1. X is totally order-disconnected. Hence, a perfect Kripke frame X = hX, ≤, τ i is an Esakia
space iff ≤ is a partial order.
2. For every F ∈ K(X), F ↑ and F ↓ ∈ K(X). Hence, for every x ∈ X, x↑ and x↓ ∈ K(X).
164
Proof. 1. If x, y ∈ X and x 6≤ y, then y ∈
/ x↑ ∈ K(X). As X is a Stone space, then y ∈ U and
U ∩ x↑ = ∅ for some clopen U ⊆ X. Then U ↓ is clopen, y ∈ U ↓ and U ↓ ∩ x↑ = ∅, which proves
total order-disconnectedness.
2. Analogous to the proof of 3.2.
Definition 7.6. (Perfect augmented Kripke frame) (cf. Section 3.1 of [2]) A perfect augmented Kripke frame is a structure X = hX, ≤, E, τ i such that the following list of conditions is
satisfied:
P1. hX, ≤, Ei is an augmented Kripke frame (hence ≤ is a partial order).
P2. XX = hX, ≤, τ i and hX, (≤ ◦ E), τ i are perfect Kripke frames.
P3. For every clopen up-set U of XX , E[U ] is clopen.
Lemma 7.7. For every augmented Kripke frame hX, ≤, Ei, and every up-set Y , E[Y ] is an up-set.
Proof. Let x ∈ E[Y ] and x ≤ z and let us show that z ∈ E[Y ]: as x ∈ E[Y ] then yEx for some
y ∈ Y , so z ≥ xEy, hence, as (≥ ◦ E) ⊆ (E ◦ ≥) by 7.3, zEv ≥ y for some v ∈ X, i.e. y ≤ vEz,
and as Y is an up-set and y ∈ Y , then v ∈ Y and so z ∈ E[Y ].
Lemma 7.8. (cf. Lemma 3.1 (1) of [2]) For every perfect augmented Kripke frame X = hX, ≤, E, τ i
and every x ∈ X, E[x] = (≤ ◦ E)[x] ∩ (E ◦ ≥)[x].
For every perfect augmented Kripke frame X = hX, ≤, E, τ i let us define GX = hX, ≤, E, Aτ i,
where Aτ is the collection of the clopen up-sets of hX, ≤, τ i. Notice that XGX = XX .
For every IK-space G = hX, ≤, E, Ai such that E is an equivalence relation let us consider its
associated space XG = hX, ≤, τ i and define XG = hX, ≤, E, τ i. Clearly, XXG = hX, ≤, τ i = XG .
Proposition 7.9.
1. For every perfect augmented Kripke frame X = hX, ≤, E, τ i, GX = hX, ≤, E, Aτ i is an
MIPC-space.
2. For every MIPC-space G = hX, ≤, E, Ai, XG = hX, ≤, E, τ i is a perfect augmented Kripke
frame.
Proof. 1. By P2, it holds that XGX = XX = hX, ≤, τ i is a perfect Kripke frame and ≤ is a partial
order, so by 7.5 (1) XGX is an Esakia space, and Aτ is the algebra of the clopen up-sets of XGX ,
hence D1 holds.
The assumption that hX, (≤ ◦ E), τ i is a perfect Kripke frame implies that: a) by 7.5 (2), for
every x ∈ X E[x↑] ∈ K(XGX ), which is D4; and b) for every U ∈ Aτ (U is a clopen up-set of
XGX , hence (X \ U ) is clopen, therefore), (≤ ◦ E)−1 [X \ U ] is clopen, hence so is its complement
X \ (≤ ◦ E)−1 [X \ U ] = 2(≤◦E) U . 2(≤◦E) U is also an up-set, for if z ∈ 2(≤◦E) U and z ≤ y, then
y↑ ⊆ z↑, and so E[y↑] ⊆ E[z↑] ⊆ U , hence y ∈ 2(≤◦E) U . Let us show that for every U ∈ Aτ ,
3E U ∈ Aτ : Since E is symmetric, then for every clopen up-set U , 3E U = E −1 [U ] = E[U ], which
is clopen by 7.6 P3, and it is also an up-set by 7.7. So this completes the proof of D2. By 7.8, for
every x ∈ X, E[x] = (≤ ◦ E)[x] ∩ (E ◦ ≥)[x] = E[x↑] ∩ E[x]↓, which is D5. Since hX, (≤ ◦ E), τ i is a
perfect Kripke frame, then by 7.5 (2), we get that for every x ∈ X, E[x]↓ = (≤◦E)−1 [x] ∈ K(XGX ).
Then, (recall that D4 holds) E[x] = E[x↑] ∩ E[x]↓ is the intersection of two closed sets, so it is
closed, which is D3.
2. By 4.2 (3) it holds in particular that (E ◦ ≤) ⊆ (≤ ◦ E), so hX, ≤, Ei is an augmented Kripke
frame, which is P1. D2 implies that for every clopen up-set U , E[U ] = 3E U is clopen, which is
P3. By D1, hX, ≤, τ i is an Esakia space, hence it is a perfect Kripke frame. So the only thing
we need to show is that hX, (≤ ◦ E), τ i is a perfect Kripke frame. Clearly, hX, (≤ ◦ E), τ i is an
ordered Stone space. By D4, it holds that (≤ ◦ E)[x] = E[x↑] ∈ K ↑ (XG ) for every x ∈ X, so we
are left to show that for every V clopen, (≤ ◦ E)−1 [V ] is clopen. For the remainder of this proof,
we rely on definitions and facts that can be found in the appendix of this section. By D4, the
assignment x 7−→ (≤ ◦ E)[x] = E[x↑] ∈ K ↑ (XG ) defines a map ρ : XG −→ K↑ (XG ) such that for
every clopen subset U of XG (≤ ◦ E)−1 [U ] = ρ−1 [m(U ) ∩ K ↑ (XG )], and since m(U ) ∩ K ↑ (XG ) is
a clopen subset of K↑ (XG ), it is enough to show that ρ is continuous. Since
165
B↑ = {t(U ) ∩ K ↑ (XG ) | U clopen up-set } ∪ {m(V ) ∩ K ↑ (XG ) | V clopen down-set}
is a subbase of K↑ (XG ), and the elements in the right-hand side of the union above are just the
complements of the elements in its left-hand side, it is enough to show that for every clopen up-set
U of XG , ρ−1 [t(U ) ∩ K ↑ (XG )] = ρ−1 [t(U )] is a clopen subset of XG . For every clopen up-set U of
XG , ρ−1 [t(U )] = {x ∈ X | (≤ ◦ E)[x] ⊆ U ]} = 2(≤◦E) U , which is a clopen up-set by D2.
Definition 7.10. (Morphism of perfect augmented Kripke frames) (cf. Section 3.1 of
[2]) Let Xi = hXi , ≤i , Ei , τi i be perfect augmented Kripke frames, i = 1, 2. A continuous map
f : hX1 , τ1 i −→ hX2 , τ2 i is a morphism iff the following list of conditions is satisfied for every
x, x0 , y ∈ X1 , z ∈ X2 :
M1. if x ≤1 y then f (x) ≤2 f (y).
M2. If f (x) ≤2 z then f (x0 ) = z for some x0 ∈ x↑.
M4’. If x(≤1 ◦ E1 )y then f (x)(≤2 ◦ E2 )f (y).
M6’. If f (x)(≤2 ◦ E2 )z then z = f (x0 ) for some x0 ∈ (≤1 ◦ E1 )[x].
M5. If f (x)E2 z then z ≤2 f (x0 ) for some x0 ∈ E1 [x].
Proposition 7.11.
1. For every morphism f : X1 −→ X2 of perfect augmented Kripke frames, f is a p-morphism
between the associated MIPC-spaces GX1 and GX2 .
2. For every p-morphism f : G1 −→ G2 of MIPC-spaces, f is a morphism between the associated
perfect augmented Kripke frames XG1 and XG2 .
Proof. 1. We have to show the conditions M3, M4 and M6 in 3.12 hold: M3 is equivalent to the
continuity of f , and M6’ immediately implies M6. Let us show M4, i.e. assume that xE1 y and
show that f (x)E2 f (y): By 7.8, it is enough to show that f (x)(≤ ◦ E2 )f (y) and f (x)(E2 ◦ ≥)f (y).
As x ≤ xE1 y, then by M4’, f (x)(≤ ◦ E2 )f (y). As xE1 y ≥ y, then x ∈ (≤ ◦ E1 )[y] so, by M4’,
f (x) ∈ (≤ ◦ E2 )[f (y)] i.e. f (y) ∈ (≤ ◦ E2 )−1 [f (x)] = (E2 ◦ ≥)[f (x)].
2. We have to show that f is continuous and that M4’, M6’ in 7.10 hold: M3 is equivalent to
continuity, and M4’ is easily implied by M1 and M4. Let us show M6’: assume that f (x)(≤2 ◦E2 )z,
and show that z = f (x0 ) for some x0 ∈ (≤1 ◦ E1 )[x]. By M6, f (y) ≤2 z for some y ∈ (≤ ◦ E1 )[x],
hence, by M2, z = f (x0 ) for some x0 ∈ y↑, and as y ∈ (≤ ◦ E1 )[x], then x0 ∈ (≤ ◦ E1 ◦ ≤)[x] =
(≤ ◦ E1 )[x], the last equality being implied by (E1 ◦ ≤) ⊆ (≤ ◦ E1 ).
Putting 7.9 and 7.11 together yields:
Theorem 7.12. MIPCSp (MIPC-spaces and their p-morphisms) and TPSOE (perfect augmented
Kripke frames and their morphisms) are isomorphic categories.
7.13
Appendix
The proofs of the facts mentioned in this appendix will appear in a forthcoming paper. Recall
that for every X = hX, τ i, the Vietoris topology (cf. [13]) τv on K(X) is the one generated by
taking the following collection as a subbase:
{t(A) | A ∈ τ } ∪ {m(A) | A ∈ τ },
where for every A ∈ τ , t(A) = {F ∈ K(X) | F ⊆ A} and m(A) = {F ∈ K(X) | F ∩ A 6= ∅}.
K(X) = hK(X), τv i is the Vietoris space of X. It is well-known that for every Stone space X,
K(X) is a Stone space, moreover for every clopen subset U of X, m(X \ U ) = K(X) \ t(U ) and
t(X \ U ) = K(X) \ m(U ), hence t(U ) and m(U ) are clopen subsets of K(X).
Proposition 7.14. Esakia spaces are exactly those Priestley spaces X = hX, ≤, τ i such that
K ↑ (X) is a closed subset of the Vietoris space of hX, τ i.
166
This implies that, for every Esakia space X = hX, ≤, τ i, the collection K ↑ (X) of the closed up-sets
of X, endowed with the inherited Vietoris topology τv0 , is a Stone space: So, for every Esakia space
X = hX, ≤, τ i, the space K↑ (X) = hK ↑ (X), ⊇, τv0 i is an ordered Stone space, that can be shown
to be also totally order-disconnected (i.e. a Priestley space). As a byproduct of the proof of this
last fact one gets:
Proposition 7.15. For every Esakia space X = hX, ≤, τ i,
B ↑ = {t(U ) ∩ K ↑ (X) | U clopen up-set } ∪ {m(V ) ∩ K ↑ (X) | V clopen down-set}
is a subbase of K↑ (X).
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